Finding Vertex And Y-Intercept Of Y=2x^2+16x+33

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Hey guys! Today, we're diving into the fascinating world of quadratic equations and their graphical representations. Specifically, we're going to tackle the problem of finding the vertex and the y-intercept of a quadratic equation in the standard form: y = ax² + bx + c. This is a fundamental skill in algebra, and mastering it will unlock a deeper understanding of parabolas and their applications. So, let's get started and make this concept crystal clear!

Understanding Quadratic Equations and Parabolas

Before we jump into the calculations, let's quickly recap what quadratic equations and parabolas are all about. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually x) is 2. The general form of a quadratic equation is y = ax² + bx + c, where a, b, and c are constants, and a is not equal to 0. The graph of a quadratic equation is a parabola, a U-shaped curve that opens either upwards or downwards, depending on the sign of the coefficient a. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards. This simple rule is your first clue in visualizing the shape of the graph.

Understanding the relationship between the coefficients (a, b, and c) and the parabola is crucial. The coefficient a not only determines the direction the parabola opens but also its width. A larger absolute value of a results in a narrower parabola, while a smaller absolute value leads to a wider one. The coefficient b influences the position of the parabola's axis of symmetry, and c directly gives us the y-intercept, which is the point where the parabola crosses the y-axis. This initial understanding sets the stage for our detailed calculations.

The vertex of the parabola is a critical point. It represents the minimum or maximum value of the quadratic function. If the parabola opens upwards (a > 0), the vertex is the minimum point, and if the parabola opens downwards (a < 0), the vertex is the maximum point. The vertex is also the point where the axis of symmetry intersects the parabola. The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. Finding the vertex is a key step in analyzing the behavior of the quadratic function and its graph. It helps us understand the range of the function, the intervals where the function is increasing or decreasing, and the overall shape of the parabola. Think of the vertex as the "turning point" of the parabola, the spot where the function changes direction.

Finding the Vertex: A Step-by-Step Approach

Now, let's get to the heart of the matter: how to find the vertex of a parabola. There are a couple of methods we can use, but the most common and generally applicable one involves using a formula derived from completing the square. Remember our equation: y = ax² + bx + c? The x-coordinate of the vertex, often denoted as h, can be found using the formula:

h = -b / 2a

This formula is your best friend when it comes to locating the vertex. It directly relates the x-coordinate of the vertex to the coefficients a and b of the quadratic equation. Once you've calculated h, you can find the y-coordinate of the vertex, often denoted as k, by substituting h back into the original equation:

k = a(h)² + b(h) + c

This two-step process is the key to unlocking the vertex. First, find the x-coordinate using the formula, and then plug that value back into the original equation to find the corresponding y-coordinate. Let's illustrate this with our specific equation: y = 2x² + 16x + 33. In this case, a = 2, b = 16, and c = 33. Applying the formula, we get:

h = -16 / (2 * 2) = -16 / 4 = -4

So, the x-coordinate of the vertex is -4. Now, we substitute this value back into the equation to find the y-coordinate:

k = 2(-4)² + 16(-4) + 33 = 2(16) - 64 + 33 = 32 - 64 + 33 = 1

Therefore, the vertex of the parabola is (-4, 1). This means the parabola turns at the point where x is -4 and y is 1. This vertex is crucial for understanding the graph's behavior, especially its minimum or maximum value.

Let's break down the process into even simpler steps:

  1. Identify a, b, and c: From the equation y = ax² + bx + c, identify the coefficients a, b, and c. These values are the foundation of our calculations. In our example, a = 2, b = 16, and c = 33.
  2. Calculate h: Use the formula h = -b / 2a to find the x-coordinate of the vertex. This step directly gives you the horizontal position of the vertex on the graph. For our equation, h = -16 / (2 * 2) = -4.
  3. Calculate k: Substitute the value of h back into the original equation y = ax² + bx + c to find the y-coordinate of the vertex. This step completes the vertex coordinates. For our equation, k = 2(-4)² + 16(-4) + 33 = 1.
  4. Write the vertex as a coordinate pair: Express the vertex as a coordinate pair (h, k). This gives you the exact location of the vertex on the coordinate plane. In our case, the vertex is (-4, 1).

By following these steps diligently, you can confidently find the vertex of any quadratic equation. Remember, practice makes perfect, so try this method with various quadratic equations to solidify your understanding.

Finding the Y-Intercept: A Simple Substitution

Now that we've conquered the vertex, let's move on to finding the y-intercept. The y-intercept is the point where the parabola intersects the y-axis. This is the point where the x-coordinate is equal to 0. Finding the y-intercept is arguably the easiest part of analyzing a quadratic equation. To find the y-intercept, we simply substitute x = 0 into the original equation and solve for y. Let's do it for our equation: y = 2x² + 16x + 33.

Substituting x = 0, we get:

y = 2(0)² + 16(0) + 33 y = 0 + 0 + 33 y = 33

Therefore, the y-intercept is (0, 33). This means the parabola crosses the y-axis at the point where y is 33. The y-intercept gives us a crucial anchor point for graphing the parabola and understanding its vertical position.

The y-intercept is often the most straightforward feature to find because it directly relates to the constant term in the quadratic equation. In the standard form y = ax² + bx + c, the y-intercept is simply the point (0, c). This direct connection makes it a quick and easy way to get a starting point for sketching the parabola.

To summarize, finding the y-intercept involves a single step:

  1. Substitute x = 0: Replace x with 0 in the original equation. This sets up the equation to solve for the y-coordinate of the y-intercept.
  2. Solve for y: Calculate the value of y. This gives you the y-coordinate of the y-intercept.
  3. Write the y-intercept as a coordinate pair: Express the y-intercept as a coordinate pair (0, y). This clearly identifies the point where the parabola intersects the y-axis.

With this method, you can easily find the y-intercept of any quadratic equation. It's a simple yet valuable tool for understanding and visualizing the parabola's position on the coordinate plane.

Putting It All Together: Our Example Solved

Let's recap our journey and put everything together for the equation y = 2x² + 16x + 33. We've systematically found both the vertex and the y-intercept, giving us a clear picture of this parabola.

First, we tackled the vertex. We identified a = 2 and b = 16, and used the formula h = -b / 2a to find the x-coordinate of the vertex:

h = -16 / (2 * 2) = -4

Next, we substituted h = -4 back into the original equation to find the y-coordinate of the vertex:

k = 2(-4)² + 16(-4) + 33 = 1

So, the vertex is (-4, 1). This point is the turning point of our parabola, and since a = 2 is positive, we know the parabola opens upwards, making this vertex the minimum point.

Then, we moved on to the y-intercept. We substituted x = 0 into the original equation:

y = 2(0)² + 16(0) + 33 = 33

Therefore, the y-intercept is (0, 33). This is where the parabola crosses the y-axis, giving us another crucial point for sketching the graph.

Combining these two key features, we now know the vertex and the y-intercept of the parabola represented by the equation y = 2x² + 16x + 33. The vertex is (-4, 1), and the y-intercept is (0, 33). This information allows us to accurately sketch the parabola and understand its behavior. We know it's a U-shaped curve that opens upwards, has its minimum point at (-4, 1), and crosses the y-axis at (0, 33). This detailed analysis showcases the power of finding the vertex and y-intercept in understanding quadratic equations.

Therefore, the correct answer from our options is:

  • a) Vertex: (-4, 1); y-intercept: (0, 33)

This comprehensive solution not only provides the correct answer but also walks through the entire process, ensuring a clear understanding of how to find the vertex and y-intercept. Remember, practice is key, so try these methods with different quadratic equations to master this fundamental skill.

Why Finding the Vertex and Y-Intercept Matters

Now that we've mastered the techniques for finding the vertex and y-intercept, let's discuss why these are so important. These two points provide crucial information about the graph of a quadratic equation, the parabola. They act as key landmarks that help us accurately sketch the graph and understand its behavior. The vertex, as we've seen, is the turning point of the parabola. It represents the minimum or maximum value of the quadratic function, depending on whether the parabola opens upwards or downwards. This is incredibly useful in real-world applications where we want to find optimal values, such as maximizing profit or minimizing costs.

The y-intercept, on the other hand, tells us where the parabola intersects the y-axis. This is the value of the function when x = 0. In many practical scenarios, the y-intercept has a meaningful interpretation. For example, if we're modeling the trajectory of a projectile, the y-intercept might represent the initial height of the projectile. Together, the vertex and y-intercept provide a solid foundation for understanding and interpreting quadratic functions and their applications.

Furthermore, finding the vertex and y-intercept helps us analyze the properties of the quadratic function. We can determine the axis of symmetry, which is a vertical line passing through the vertex that divides the parabola into two symmetrical halves. We can also identify the range of the function, which is the set of all possible y-values. These properties are essential for solving quadratic equations, graphing parabolas, and applying quadratic functions to real-world problems.

The ability to find the vertex and y-intercept is a gateway to more advanced topics in algebra and calculus. These concepts are used extensively in optimization problems, curve sketching, and modeling real-world phenomena. By mastering these fundamental skills, you'll be well-equipped to tackle more complex mathematical challenges.

In conclusion, finding the vertex and y-intercept is not just a mathematical exercise; it's a powerful tool for understanding and applying quadratic functions. These points provide crucial information about the graph of the parabola, allowing us to sketch it accurately, analyze its properties, and solve real-world problems. So, keep practicing, keep exploring, and you'll unlock the full potential of quadratic equations!